The Calculus of Differentials for the Weak Stratonovich Integral

Abstract

The weak Stratonovich integral is defined as the limit, in law, of Stratonovich-type symmetric Riemann sums. We derive an explicit expression for the weak Stratonovich integral of f (B) with respect to g(B), where B is a fractional Brownian motion with Hurst parameter 1/6, and f and g are smooth functions. We use this expression to derive an Itô-type formula for this integral. As in the case where g is the identity, the Itô-type formula has a correction term which is a classical Itô integral, and which is related to the so-called signed cubic variation of g(B). Finally, we derive a surprising formula for calculating with differentials. We show that if dM = X dN , then Z dM can be written as ZX dN minus a stochastic correction term which is again related to the signed cubic variation.

Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion

“…The signed cubic variation of B is defined in [8] as a class of sequences of processes, each of which is equivalent, in a certain sense, to the sequence {W n }. The relevant fact for our present purposes is that the sequence {W n } converges in law to a Brownian motion independent of B.…”

“…In the particular case k = 3, the variance, denoted by κ 2 t, is given in formula (2.1) below. A detailed analysis of the signed cubic variation of B has been recently developed by Swanson in [8], considering this variation as a class of sequences of processes.…”

Stratonovich-type integral with respect to a general stochastic measure

“…But in some cases, such as stochastic integration on manifolds or Wong-Zakai approximation of stochastic equations, Stratonovich integral is a very useful tool. Stratonovich-type definitions of stochastic integrals with respect to differrent classes of processes may by founded in [9], [3], [6], and [19].…”

Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion II

“…The process W is related to the signed cubic variation of B. A detailed analysis of this process has been recently developed by Swanson in [8], considering this variation as a class of sequences of processes.…”